Representation of ₄-Linear Preparata Codes Using Vector Fields

ISSN： 0032-9460

Source： Problems of Information Transmission, Vol.41, Iss.2, 2005-04, pp. : 113-124

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Abstract

A binary code is called ₄-linear if its quaternary Gray map preimage is linear. We show that the set of all quaternary linear Preparata codes of length n = 2^m, m odd, m ≥ 3, is nothing more than the set of codes of the form $$\mathcal{H}_{\lambda ,\not \upsilon } + \mathcal{M}$$ with $$\mathcal{H}_{\lambda ,\not \upsilon } = \{ y + T_\lambda (y) + S_{\not \upsilon } (y)|y \in H^n \} ,\quad \mathcal{M} = 2H^n ,$$ where T _λ(⋅) and S _ψ (⋅) are vector fields of a special form defined over the binary extended linear Hamming code H ⁿ of length n. An upper bound on the number of nonequivalent quaternary linear Preparata codes of length n is obtained, namely, $$2^{n\log _2 n}$$ . A representation for binary Preparata codes contained in perfect Vasil’ev codes is suggested.

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